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Morfill 1975
Morfill, G. (1975). A two-component description of energetic particle scattering in a turbulent magnetoactive plasma. Journal of Geophysical Research 80: doi: 10.1029/JA080i013p01783. issn: 0148-0227.

The problem of resonant scattering of charged particles in a turbulent plasma is described in terms of a simplified diffusion coefficient which allows a good quantitative estimate to be made of D&mgr; (pitch angle diffusion coefficient) and k (spatial diffusion coefficient) for a general case of a wave k vector distribution, which is anisotropic and not necessarily field-aligned. Since the diffusion coefficient depends very strongly on the spectral form (i.e., k vector distribution) of the power in the magnetic fluctuations, it is important to be able to quantify k for such a general distribution of k vectors, in order to make meaningful comparison with observations. The simplified expression derived here for the diffusion coefficient takes advantage of the fact that the quasi-linear theory gives and analytic expression in all those cases where the wave k vector distribution is a delta function. The full k vector distribution is represented as the sum of two weighted delta function approximations, one of which is parallel to the average magnetic field B0 and the other parallel to the direction of maximum k vector density A. The component parallel to B0 turns out to be very important when the k vector distribution is very asymmetric, i.e., the angle between A and B0 is large. The simplified expression for the pitch angle diffusion coefficient is D&mgr;= (q/&ggr;mc)2(1-&mgr;2) {1/8gBP (fB) fB&ggr;d&ohgr;B/&OHgr; +1/2gA&Sgr;n=1P (nfA) Jn2[n (1-&mgr;2)1/2 tan &khgr;/&mgr;>&mgr;2FA&ggr; (2&pgr;-d&ohgr;B)/(1-&mgr;2)tan2&khgr; &OHgr;}, where q, m, c, &ggr;, &OHgr;/&ggr;, and &mgr; are the particle charge, mass, velocity of light, dimensionless particle energy, gyrofrequency, and cosine of particle pitch angle, respectively. In principle, one can use any functional dependence of the k vector density distribution g (ϑ,ϕ), but for illustration purposes this paper uses the simplest form, i.e., g (ϑ,ϕ) =[1/&pgr; (C+2)>(1+C sin ϑ sin ϕ). The axis of anisotropy A is at an angle &khgr; to the magnetic field direction, and thus gB= (1+C cos &khgr;)/&pgr; (C+2) and gA= (1+C)/&pgr; (C+2). Here C?1 expresses the size of the anisotropy. The terms fb and fa are the Doppler-shifted resonance frequencies along the axes A and B. Thus fB= (&OHgr;/2&pgr;&ggr;w&mgr;)[(vvA+vA2) /‖vA‖>, and fa= (&OHgr;/2&pgr;&ggr;w&mgr;) [(vA+‖vA‖cos &khgr;>, where vA is the Alfv¿en velocity, v the plasma velocity, and w the particle velocity. The term P (f) is the power spectrum of the magnetic fluctuations at frequency f, d&ohgr;B

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Journal of Geophysical Research
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